Sparse equidistribution problems in dynamics

26 February - 28 May 2025

Wednesdays, 10:15 - 12:00

Location: HG G 43

First lecture: 26 February

Abstract

Let \((X,T)\) be a topological dynamical system. Classical ergodic theory studies asymptotic behavior of averages of the form \(\frac{1}{N}\sum_{n\leq N}f(T^nx)\) where \(x\in X\) and \(f\in C(X)\). There are many fundamental results on convergence of such averages in various norms: the supremum norm, almost everywhere and in \(L^2\) being the main examples (in the latter two we equip \((X,T)\) with a measure \(\mu\) which is preserved by \(T\)). Recently there has been a lot of interest in studying non-classical averages. In the most general form, given a bounded sequence \((a_n)\) one is interested in the asymptotic behavior of averages:

\(\frac{1}{\sum_{n\leq N}|a_n|}\sum_{n\leq N}a_n\cdot f(T^nx)\).

In most applications the sequence \((a_n)\) is a structured sequence coming from number theory. Some of the examples of such sequences include:

1. \(a_n\)= Möbius function or Liouville function (or more generally a multiplicative function).
2. for \(k\geq 1\), \(a_n=1\) if \(n\) is a product of at most \(k\) primes, \(a_n= 0\) otherwise.
3. \(a_n=1\) if \(n\) is a k-th power (like square) and \(a_n=0\) otherwise.
4. \((a_n)\) a random sequence of real numbers.